First amendment protections from censorship are exclusive to addressing government actions, but censorship itself doesn't require government action.
Social media can (and should be allowed to) ban who they want on their own platforms as long as those platforms ain't owned by the state. That doesn't mean there are no problems with a culture of censorship or that such bans can't be a net negative to society. It doesn't mean that the chilling effect of deplatforming fear doesn't make us less open with one another and it doesn't mean that echo chambers aren't still encouraged by the system. It merely means the solution to any of these problems shouldn't be a legal one.
Josh Hawley is wrong for the same reason Andrew Yang is wrong. They're correctly, importantly, highlighting serious problems that should be addressed but proposing the wrong solutions.